Homomorphic encryption is a form of encryption where a specific algebraic operation (e.g., addition or multiplication) performed on the plaintext can be equivalent to another (possibly different) algebraic operation performed on the ciphertext. An encryption scheme which supports both addition and multiplication (thereby preserving the ring structure of plaintexts) is generally known as fully homomorphic encryption (FHE).
Theoretically, FHE can allow any function, including those computed by miscellaneous complex operations (consisting of additions, multiplications, etc.) homomorphically, e.g., encrypted queries to a search engine, or searching on encrypted data (in a database). Such a scheme would allow computation of arbitrary functions over encrypted data without having to perform any decryption during this computation.
As those skilled in the art readily appreciate, techniques for encryption, especially homomorphic encryption are highly useful.